A347926 Decimal expansion of the smallest a such that log(1 + x) <= x^a for all x >= 0.

3, 7, 9, 8, 3, 1, 2, 1, 4, 9, 2, 6, 6, 1, 0, 9, 0, 1, 8, 2, 2, 6, 1, 0, 0, 5, 6, 7, 2, 1, 2, 2, 9, 2, 4, 4, 1, 7, 6, 2, 9, 1, 0, 7, 2, 5, 8, 6, 3, 9, 1, 5, 3, 3, 5, 4, 8, 1, 5, 6, 5, 5, 5, 7, 7, 6, 8, 2, 7, 1, 7, 4, 5, 2, 5, 2, 0, 6, 3, 8, 8, 9, 0, 8, 4, 7, 3, 7, 9, 8, 0, 8, 8, 7, 3, 3, 4, 7, 5, 8, 2, 2, 8, 1, 3

Offset

0,1

Comments

Fredrik Johansson remarks: "The inequality log(1 + x) <= x is used all the time. Putting x^a on the right gives a bound that grows less quickly and which remains easy to manipulate multiplicatively."

Links

Table of n, a(n) for n=0..104.

Example

0.37983121492661090182261...

Maple

Digits := 120: Optimization:-Maximize(log(log(1 + x))/log(x), {x>=9})[1]:
evalf(%)*10^105: ListTools:-Reverse(convert(floor(%), base, 10));

Mathematica

RealDigits[FindMaximum[Log[Log[1 + x]]/Log[x], {x, 7}, WorkingPrecision -> 110][[1]], 10, 105][[1]] (* Amiram Eldar, Oct 16 2021 *)

Crossrefs

Sequence in context: A275667 A122490 A365317 * A331066 A179021 A096910

Adjacent sequences: A347923 A347924 A347925 * A347927 A347928 A347929

Keyword

nonn,cons

Author

Peter Luschny, Oct 16 2021

Status

approved